Parent guide · 6 min read
Box plots for kids: five numbers, four quarters, and the box that is not the whole story
A box plot (also called a box and whisker plot) squeezes a whole list of numbers into five key values and one small picture. It is compact, which is why it is useful, and also why it confuses children: there are no individual numbers to see, only a box, a line inside it and two whiskers. This guide explains the five numbers, shows how to find them from a list, and clears up the most common misreading, which is thinking that a longer section means more data.
Try this tonight
5 things you can do at home
Sort the list first
Put the numbers in order from smallest to largest before doing anything else. Every one of the five values is read from the sorted list, and most box plot errors begin with an unsorted one.
Find the median, then split the list in two
The median is the middle value. If there is an odd number of values, set the median aside; the values below it are the lower half and the values above it are the upper half. If there is an even number, split the list straight down the middle.
Find the middle of each half
The median of the lower half is the first quartile (Q1) and the median of the upper half is the third quartile (Q3). Say it out loud: "the middle of the bottom half" and "the middle of the top half".
Mark all five numbers on a number line
Draw one number line with equal spacing, mark the smallest value, Q1, the median, Q3 and the largest value, then draw the box from Q1 to Q3 with a line at the median and whiskers out to the ends.
Describe it in a sentence
Finish with something like "Half the scores are between ___ and ___, and the middle score is ___." If the child can say that, they have read the plot, not just drawn it.
What the five numbers mean
The minimum and maximum are the smallest and largest values. The median is the middle value, with half the data below it and half above. The first quartile has a quarter of the data below it, and the third quartile has a quarter of the data above it. Together they cut the sorted list into four parts, each holding about the same number of values.
The box runs from Q1 to Q3, so it covers the middle half of the data. Its length, Q3 minus Q1, is called the interquartile range (IQR). It is a measure of how spread out the middle of the data is, and unlike the full range it is not stretched by one very high or very low value.
A worked example: 11 test scores
Eleven students scored 52, 60, 64, 68, 70, 72, 75, 78, 82, 90 and 96 on a test, already in order. The median is the 6th value, which is 72. Setting it aside, the lower half is 52, 60, 64, 68, 70, whose middle value is 64, so Q1 = 64. The upper half is 75, 78, 82, 90, 96, whose middle value is 82, so Q3 = 82.
The five-number summary is 52, 64, 72, 82, 96. The box runs from 64 to 82, so the IQR is 82 − 64 = 18, and the full range is 96 − 52 = 44. A good sentence to finish with: "Half the students scored between 64 and 82, and the middle score was 72."
The trap: a longer section does not mean more data
In the example, the part of the box from 72 to 82 is longer than the part from 64 to 72, and the right whisker (82 to 96) is longer than the left (52 to 64). It is tempting to read the longer sections as holding more students. They do not: each of the four sections holds about a quarter of the data.
What a long section tells you is that those values are more spread out. The scores in the top quarter ran from 82 to 96, a gap of 14 points, while the scores in the bottom quarter ran from 52 to 64, a gap of 12. Reading length as spread, and not as count, is the main idea to hold on to.
Comparing two box plots
Box plots are most useful side by side on one number line. Suppose another class has a median of 72 but a box from 68 to 76. The middle scores are the same, but the second class has an IQR of 8 against 18, so its middle half is far more tightly packed.
That is something the median alone cannot show, and it is why the box plot is taught after the median. When two sets of data have the same centre, the spread is often where the difference is.
Signs it’s time for outside help
When home help has done what it can
- Finds the median of the unsorted list.
- Includes the median in both halves when the list has an odd number of values.
- Thinks the whiskers or the box show every value.
- Reads a longer section as meaning more data in it.
- Confuses the range with the interquartile range.
- Draws the five numbers on an unevenly spaced number line.
Questions parents ask
FAQ
What grade are box plots taught?
In the US Common Core, box plots appear in Grade 6 statistics (ages 11–12) and are used again in Grade 7 to compare two sets of data. Other curricula, including the Australian one, cover them across the early secondary years.
What is the five-number summary?
The smallest value, the first quartile, the median, the third quartile and the largest value. A box plot is a picture of those five numbers.
What is the interquartile range?
It is the third quartile minus the first quartile, which is the length of the box. It measures the spread of the middle half of the data.
Do all methods give the same quartiles?
Not always. For an odd number of values, some methods include the median in both halves and others leave it out, which can shift Q1 and Q3 slightly. It is best to use the method your child’s class or textbook uses; this guide leaves the median out.
Can you see the mean on a box plot?
No. A box plot shows the median and the quartiles, not the mean, and it does not show individual values.
Further reading
Classes that fit
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